Download 1. Introduction 2. Minimum, Maximum, Range, Percentiles and the IQR

Statistics 10
1.
Lecture 8
Percentiles & Transformations (5.4-5.6)
Introduction
While many variables have distributions that are approximately normal, there are just as many situations when the
distribution is not well represented by the normal curve. Recall that the median is an alternative way to describe
CENTER and so there are alternative ways to describe SPREAD.
2.
Minimum, Maximum, Range, Percentiles and the IQR (Chapter 5.4)
A. The lowest and highest values of a variable are the minimum and maximum.
B. The range is maximum-minimum.
C. Definition: a number y is the nth PERCENTILE for the data if n% of the data are less than or equal to y.
D. The QUARTILES are the 25th, 50th, and 75th percentiles and are denoted Q1, Q2, and Q3. (Another name
for Q2 is the median.)
E. The INTER-QUARTILE RANGE, or IQR, is defined as IQR = (the value of Q3 – the value of Q1); the IQR
measures the spread of the middle 50% of the data. It is used in some cases as a replacement for the SD.
3.
Percentiles and the Normal Curve (Chapter 5.5)
Standard (Z) scores vs. percentiles: If all one does with standard scores is convert them to percentiles, then why have
both?
Percentiles and standard scores have slightly different information in them. In other words, the move from standard
scores to their normal curve percentile equivalents is not a smooth one-to-one transformation. Percentiles shows how a
score relates to other scores in a distribution: it is the value below which the proportion of the cases in the distribution
falls. On the other hand, a standard score (Z) can be used to answer a few more questions:
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•
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What percentage fall below a given standard score?
What is the relative standing of a standard score in one distribution versus another distribution?
What score or scores can be used to define an extreme or deviant situation?
Very large differences between standard scores far away from the mean (in either direction) correspond to small
differences in percentiles; likewise, very small differences in standard scores near the mean correspond to large
differences in percentiles.
4.
Changing Scales (Chapter 5.6)
Definition: changing the original values in a list of numbers by adding, subtracting, multiplying or dividing each value
by a constant. All a Z score really does is change the scale of an existing set of values by subtracting the mean from a
value and then dividing by the standard deviation. What effect does this have? It affects both “center” and “spread”.
The mean will be changed (added to, subtracted from, multiplied or divided by the constants we used)
The standard deviation will be changed (multiplied by or divided by the constants we used)
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Essentially, this is how the standard normal curve in Table A-105 works. Every set of numbers converted to
Standard Units will have a mean of 0 and a SD=1.
•
The values of the mean, median, the minimum, maximum, percentiles, are affected by both the addition (or
subtraction) of a constant and multiplication (or division) by a constant.
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The values of the measures of spread – that is, the IQR and the standard deviation, and the range, are only
affected by multiplication (or division) by a constant.
Statistics 10
5.
Lecture 8
Last words on Using the Normal Curve
A. IF the data are normally
distributed, then raw scores can be
converted into standard units to find
percentages; also, percentages can be
converted into standard units and
then converted into raw scores.
B. If the data are NOT normally
distributed, then using the normal
curve will give the wrong answer!
NEVER ASSUME THAT DATA
ARE NORMAL.
Percentiles & Transformations (5.4-5.6)